Promoting Mathematical Thinking From Worked Examples via Key

Promoting Mathematical Thinking From Worked Examples via Key ideas, Core Awarenesses and Variation to the issue of Student Attention The Open University Maths Dept 1 John Mason Open University February 2017 University of Oxford Dept of Education

Conjectures v Everything said here today is a conjecture … to be tested in your experience v The best way to sensitise yourself to learners … is to experience parallel phenomena yourself So, what you get from this session is mostly what you notice happening inside you! v 2

Outline v v 3 Worked Examples Key Ideas & Core Awarenesses Variation & Mastery Student Attention

Rhind Mathematical Papyrus problem 28 Two-thirds is to be added. One-third is to be subtracted. There remains 10. Make 1/10 of this, there becomes 1. The remainder is 9. 2/3 of this is to be added. The total is 15. 1/3 of this is 5. Lo! 5 is that which goes out, And the remainder is 10. The doing as it occurs! 4

Worked Examples v v v A 4000 year old practice “Thus is it done”; “Do it like this”; … What is the student supposed to do? – Follow the sequence of acts – Distinguish structural relations and values from parameters v v Templating What does the student need from a worked example? – What to do ‘next’ – How to know what to do next 5 Chi & Bassok Renkl & Sweller Watson & Mason

Integration Key ideas? Core Awarenesse s 6

Another Integration Key ideas? Core Awarenesse s 7

Exercises for developing facility Varying type for deepening comprehension Varying structure Doing & Undoing for extending appreciation Varying context Complexifying & Embedding for enriching accessible example spaces 8 Learners generating own examples subject to constraints Extending, & Restricting Generalising & Abstracting

Two Areas v v v 10 What is the relation between the areas of the red and blue regions? How do you know? What can be varied?

Brown & Blue Areas … how related? 11

Carpet Theorem v v 12 Static: If two carpets exactly cover a floor without overlap, then when they are shifted so there is some overlap, the area of the overlap is the same as the area of the exposed floor. Dynamic: if two carpets on the floor have their positions shifted, the change in area of overlap equals the change in area of exposed floor.

Brown Area = Blue Area What are the radii? 13

√ 2 is irrational Consider … … the smallest integer square which is twice the area of an integer-sided square Put two copies of the halfsquare into opposite corners By the carpet theorem we have another such square. . . which is also integer-sided Contradiction! There can be no such smallest integer-sided square and hence no such integer-sided square 14

Ride & Tie Two people have but one horse for a journey. One rides while the other walks. The first then ties the horse and walks on. The second takes over riding the horse … They want to arrive together at their destination. Imagine and sketch 15

Linearity in vector Spaces T maps e 1 to f 1 and e 2 to f 2 How do you find T(V)? 16

Change of Basis 17

Expressing What is Seen Sketch what you saw 18 What is the same and whatisdifferent What freedom available? about the two What can be varied? diagrams?

Variation Invariance in the midst of change Gothenburg Variation Shanghai (Confucian) Variation 19 Theorems What can change and yet something remains invariant Existence theorems Something is available to be learned only when critical aspects have been varied in local time and space Same problem different methods Same method different problems Same concept different contexts “Nettle of a Concept”

Attention v v What are students attending to? What do they need to attend to? How are they attending to it? What am I attending to? How am I attending to it? Conjecture: if students and tutor are attending to different things OR if attending to the same thing but differently THEN communication is likely to be impoverished 20

Forms of Attention v v v 21 Holding a Whole (gazing) Discerning Details Recognising Relationships in the particular situation Perceiving Properties as being instantiated Reasoning on the basis of agreed properties

Inner & Outer Aspects v Outer – What task actually initiates explicitly v Inner – – 22 What mathematical concepts underpinned What mathematical themes encountered What mathematical powers invoked What personal propensities brought to awareness

Crossed Ladders v 23 In an alleyway there is a ladder from the base of one wall to the opposite wall, and another the other way, reaching to heights 3 m and 4 m respectively on the opposite walls. Find the height of the crossing point.

Crossed Ladders Solution Dimensions of possible variation Harmonic or Parallel Sum 24

Journeys v Couriers – A courier sets out from one town to go to another at a certain pace; a few hours later the message is countermanded so a second courier is sent out at a faster pace … will the second overtake the first in time? v Meeting Point – Some people leave town A headed for town B and at the same time some people leave town B headed for town A. They all meet at noon, eating lunch together for an hour. They continue their journeys. One group reaches their destination at 7: 15 pm, while the other group gets to their destination at 5 pm. When did they all start? [Arnold] 25

Meeting Point Resolution v Draw a graph! B Distance from A A 26 Dimensions of possible variation? time

Cistern Filling v A cistern is filled by two spouts, which can fill the cistern in a and b hours respectively working alone. How long does it take with both working together? a Dimensions of possible variation? b time 27

Crossed Planes v v Imagine three towers not on a straight line. Imagine a plane through the base of two towers and the top of the third; and the other two similar planes. – They meet in a point. v Imagine a plane through the tops of two towers and the base of the third; and the other two similar planes – They meet in a point v 28 The first is the mid-point between the ground and the second.

Tower Diagrams 29

Powers & Themes Are students being encouraged to use their own powers? Powers or are their powers being usurped by textbook, worksheets and … ? Imagining & Expressing Specialising & Generalising Conjecturing & Convincing (Re)-Presenting in different modes Organising & Characterising Themes Doing & Undoing Invariance in the midst of change Freedom & Constraint Restricting & Extending Exchanging 30

Reflection as Self-Explanation v v 31 What struck you during this session? What for you were the main points (cognition)? What were the dominant emotions evoked? (affect)? What actions might you want to pursue further? (Awareness)

Frameworks Enactive – Iconic – Symbolic Doing – Talking – Recording See – Experience – Master Concrete – Pictorial– Symbolic 32

Reflection v It is not the task that is rich … but the way the task is used v v Teachers can guide and direct learner attention What are teachers attending to? … Powers … themes … Heuristics … the focus and form of their own attention 33

To Follow Up v v pmtheta. com john. mason@open. ac. uk v v 34 Thinking Mathematically (new edition 2010) Developing Thinking in Algebra (Sage) Designing & Using Mathematical Tasks (Tarquin) Questions and Prompts (ATM)